Semimartingale decomposition of convex functions of continuous semimartingales by Brownian perturbation

Details Semimartingale decomposition of convex functions of continuous semimartingales by Brownian perturbation Semimartingale decomposition of convex functions of continuous semimartingales by Brownian perturbation

2009-07-02

arxiv.org/abs/0907.0382v5

Mathematics

Probability

16 pages. Re-submitted to ESAIMPS December, 2010

Scientific paper

In this note we prove that the local martingale part of a convex function f of a d-dimensional semimartingale X = M + A can be written in terms of an It^o stochastic integral \int H(X)dM, where H(x) is some particular measurable choice of subgradient of f at x, and M is the martingale part of X. This result was first proved by Bouleau in [2]. Here we present a new treatment of the problem. We first prove the result for X' = X + eB, e > 0, where B is a standard Brownian motion, and then pass to the limit as e tends to 0, using results in [1] and [4].

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